A089058 Each positive number x is repeated m times, where there are m numbers y >= x such that x*y/(x+y) is an integer.

2, 3, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22

Offset: 1

Author

Cino Hilliard, Dec 02 2003, corrected Nov 23 2006

Keywords

• nonn

Comments

xy/(x+y) is the side of the inscribed square of a right triangle with integer legs x and y.

The number of repetitions of x is the number of divisors of x^2 that are at most x/2. - Robert Israel, May 25 2025

Examples

			For x = 6 there are 3 possible y, namely 6, 12 and 30, with 6*6/(6+6) = 3, 6*12/(6+12) = 4 and 6*30/(6+30) = 5, so 6 occurs 3 times in the sequence. - _Robert Israel_, May 28 2025
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10010

Programs

Maple

f:= n -> nops(select(`<=`, numtheory:-divisors(n^2),n/2)):
seq(n$f(n), n=1..30)]; # Robert Israel, May 25 2025

PARI

xydivxpy(n) = { for(x=1,n, for(y=x,n, h=x*y/(x+y); if(h==floor(h), print1(x","); ) ) ) }

Extensions

Definition corrected by Robert Israel, May 25 2025